How do I sperate an object while having both objects retain the same face?
Select the faces you want, then duplicate them with Shift+D. Offset in the z-direction if you want them to stay lined up (Shift+D, Z, move mouse, left-click to make it stick). Once you've done that, you'll have something that looks like this:



From here you can separate the selected faces from the object, then go down to the original and delete the faces you don't want. (Another option is to duplicate the object and delete the unwanted faces in each.)